Problem 42 - Intermediate Algebra

Question

In the following exercises, simplify the following expressions by combining like terms. $$ 5 b^{2}+9 b+10+2 b^{2}+3 b-4 $$

Step-by-Step Solution

Answer

7b^{2} + 12b + 6

Step 1: Identify like terms

First, identify all the like terms present in the expression. Like terms are those that have the same variable raised to the same power. The expression given is: 5b^{2}+9b+10+2b^{2}+3b-4

Step 2: Group like terms together

Next, group all the like terms together: (5b^{2} + 2b^{2}) + (9b + 3b) + (10 - 4)

Step 3: Combine like terms

Now, add the coefficients of each group: (5 + 2)b^{2} + (9 + 3)b + (10 - 4)

Step 4: Simplify each group

Simplify each group by performing the addition: 7b^{2} + 12b + 6

Key Concepts

simplifying expressions

When we talk about simplifying expressions, we aim to make a mathematical expression easier to work with. This usually involves combining like terms to create a more streamlined version of the original expression. For example, in the expression 5b^{2} + 9b + 10 + 2b^{2} + 3b - 4you can see that it's quite long. To simplify it, we identify terms that can be combined and then group them together, which results in fewer, more manageable terms.

like terms

Like terms are a critical concept in algebra. These are terms that have the same variable raised to the same power. In our example, the terms 5b^{2}and 2b^{2}are like terms because they both contain b^{2}. Similarly, 9band 3bare like terms because they both involve b. Constants such as 10 and -4 are also like terms because they are just numbers without any variables. By identifying and combining these like terms, you can simplify expressions much more easily.

coefficients

Coefficients are the numbers in front of the variables in algebraic terms. They tell you how many units of each variable you have. In the expression 5b^{2} + 9b + 10 + 2b^{2} + 3b - 4,the coefficients are 5, 9, 10, etc. When you're combining like terms, you're essentially adding or subtracting these coefficients. For instance, 5 and 2 are the coefficients of b^{2},so when you combine them, you get (5 + 2)b^{2}.In the same way, adding the coefficients 9 and 3 for bgives you (9 + 3)b.

algebraic operations

Algebraic operations involve adding, subtracting, multiplying, and dividing terms. In our example, we primarily focus on addition and subtraction. By combining like terms, you perform addition and subtraction operations on their coefficients. This helps in simplifying the given expression. For a simple addition, you just add the coefficients: (5 + 2)b^{2}which equals 7b^{2}.Similarly, for subtraction like (10 - 4),you simply subtract the number 4 from 10, resulting in 6.Understanding algebraic operations is essential for mastering higher-level math.